Ukrainian Mathematical Journal

, Volume 52, Issue 12, pp 1841–1857 | Cite as

Asymptotic Discontinuity of Smooth Solutions of Nonlinear q-Difference Equations

  • G. A. Derfel'
  • E. Yu. Romanenko
  • A. N. Sharkovsky


We investigate the asymptotic behavior of solutions of the simplest nonlinear q-difference equations having the form x(qt+ 1) = f(x(t)), q> 1, tR+. The study is based on a comparison of these equations with the difference equations x(t+ 1) = f(x(t)), tR+. It is shown that, for “not very large” q> 1, the solutions of the q-difference equation inherit the asymptotic properties of the solutions of the corresponding difference equation; in particular, we obtain an upper bound for the values of the parameter qfor which smooth bounded solutions that possess the property \(\begin{array}{*{20}c} {\max } \\ {t \in [0,T]} \\ \end{array} \left| {x'(t)} \right| \to \infty \)as T→ ∞ and tend to discontinuous upper-semicontinuous functions in the Hausdorff metric for graphs are typical of the q-difference equation.


Asymptotic Behavior Difference Equation Asymptotic Property Smooth Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. M. Jackson, “q-Difference equations,” Amer. J. Math., 32, 305–314 (1910).Google Scholar
  2. 2.
    R. D. Carmichael, “The general theory of linear q-difference equations,” Amer. J. Math., 34, 147–168 (1912).Google Scholar
  3. 3.
    G. D. Birkhoff, “The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations,” Proc. Amer. Acad. Arts Sci., 49, 521–568 (1913).Google Scholar
  4. 4.
    A. N. Sharkovsky, Yu. L. Maistrenko, and E. Yu. Romanenko, Difference Equations and Their Applications, Naukova Dumka, Kiev (1986). English translation:Kluwer, Dordrecht (1993).Google Scholar
  5. 5.
    E. Yu. Romanenko and A. N. Sharkovsky, “From one-dimensional to infinite-dimensional dynamical systems: ideal turbulence,” Ukr. Mat. Zh., 48, No.12, 1604–1627 (1996).Google Scholar
  6. 6.
    E. Yu. Romanenko, “On attractors of continuous difference equations,” Comp. Math. Appl., 36, No.10–12, 377–390 (1998).Google Scholar
  7. 7.
    K. Kuratowski, Topology, Academic Press, New York (1968).Google Scholar
  8. 8.
    A. N. Sharkovsky, “Coexistence of cycles of a continuous transformation of a straight line into itself,” Ukr. Mat. Zh., 16, No.1, 61–71 (1964). English translation: Int. J. Bifurc. Chaos 5, No. 5, 1263–1273 (1995).Google Scholar
  9. 9.
    A. N. Sharkovsky, “Characterization of cosine,” Aequat. Math., 9, No.2, 121–128 (1973).Google Scholar
  10. 10.
    G. P. Pelyukh and A. N. Sharkovsky, Introduction to the Theory of Functional Equations[in Russian], Naukova Dumka, Kiev (1974).Google Scholar
  11. 11.
    G. Derfel, E. Romanenko, and A. Sharkovsky, “Long time properties of solutions for simplest q-difference equations,” J. Different. Equat. Appl., 6, No.5, 485–511 (2000).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • G. A. Derfel'
    • 1
  • E. Yu. Romanenko
    • 2
  • A. N. Sharkovsky
    • 2
  1. 1.Ben-Gurion UniversityBeershebaIsrael
  2. 2.Institute of MathematicsUkrainian Academy of SciencesKiev

Personalised recommendations